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Properties

Label 7T5
7T5 1 2 1->2 1->2 3 2->3 4 3->4 6 3->6 5 4->5 5->6 7 6->7 7->1
Degree $7$
Order $168$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\GL(3,2)$

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(7, 5);
 
Copy content sage:G = TransitiveGroup(7, 5)
 
Copy content oscar:G = transitive_group(7, 5)
 

Group invariants

Abstract group:  $\GL(3,2)$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $168=2^{3} \cdot 3 \cdot 7$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Solvable:  no
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 

Group action invariants

Degree $n$:  $7$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Transitive number $t$:  $5$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
CHM label:   $L(7) = L(3,2)$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(7).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(7), G)[1])
 
Generators:  $(1,2)(3,6)$, $(1,2,3,4,5,6,7)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

7T5, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{7}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{3}$ $21$ $2$ $2$ $(2,7)(3,4)$
3A $3^{2},1$ $56$ $3$ $4$ $(2,7,6)(3,5,4)$
4A $4,2,1$ $42$ $4$ $4$ $(1,5)(2,3,7,4)$
7A1 $7$ $24$ $7$ $6$ $(1,3,4,7,6,5,2)$
7A-1 $7$ $24$ $7$ $6$ $(1,2,5,6,7,4,3)$

Malle's constant $a(G)$:     $1/2$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 

Character table

1A 2A 3A 4A 7A1 7A-1
Size 1 21 56 42 24 24
2 P 1A 1A 3A 2A 7A1 7A-1
3 P 1A 2A 1A 4A 7A-1 7A1
7 P 1A 2A 3A 4A 1A 1A
Type
168.42.1a R 1 1 1 1 1 1
168.42.3a1 C 3 −1 0 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
168.42.3a2 C 3 −1 0 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
168.42.6a R 6 2 0 0 −1 −1
168.42.7a R 7 −1 1 −1 0 0
168.42.8a R 8 0 −1 0 1 1

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 

Regular extensions

$f_{ 1 } =$ $x^{7} - 7 x^{5} + \left(t + 14\right) x^{3} + 2 t x^{2} - 14 x - 8$ Copy content Toggle raw display

Additional information

This is the lowest degree transitive permutation group which admits a Gassmann triple. As a result, number fields with this Galois group come in arithmetically equivalent pairs.